ReLu, Sum and Convolution Layers to Count Pixels of Certain Color
I can judge from the excerpt only, but I don't see any mention of a convolutional layer there. This doesn't mean that convolution isn't suitable for this task (in fact, it's the best method for image classification so far), it's just seems the instructor proposes this particular network architecture, without convolutions.

Since this architecture is more illustrative and by no means a standard, I can't be sure I fully understand it. But it seems like the first layer outputs a vector of size 4, which contains the counts of four selected colors. The ReLu operation here is a fancy way of filtering extra colors, based on channel value (note that they are close to either 0 or 255 in all four colors). If you select different four colors, this filtering will have to be done differently too, so it's basically hardwired. The second layer is more straightforward.

In general, you shouldn't consider this a "true" deep network, but merely a puzzle to get familiar with ReLu operation.